Utility Function Exercises
(Gold futures) The current price of gold is $412 per ounce The storage cost is $2 per ounce per year, payable quarterly in advance. Assuming a constant interest rate of 9% compounded quarterly, what is the theoretical forward price of gold for delivery in 9 months?
2. (Proportional carrying charges o) Suppose that a forward contract on an asset is written at time zero and there are M periods until delivery Suppose that the carrying charge in period k is qS(k), where S{k) is the spot price of the asset in period k Show that the forward price is
[Hint Consider a portfolio that pays all carrying costs by selling a fraction of the asset as required Let the number of units of the asset held at time k be x(k) and find ,\(M) in terms of .v(0) 
3. (Silver contract) At the beginning of April one year, the silver forward prices (in cents per troy ounce) were as follows:
Apr 
406.50 
July 
416 64 
Sept 
423.48 
Dec 
433.84 
(Assume that contracts settle at the end of the given month.) The carrying cost of silver is about 20 cents per ounce per year, paid at the beginning of each month Estimate the interest rate at that time
4. (Continuouslime carrying charges) Suppose that a continuoustime compounding framework is used with a fixed interest rate r Suppose thai the carrying charge per unit of time is proportional to the spot price; that is, the charge is qS(t) Show that the theoretical forward price of a contract with delivery date 7 is
jHint' Use Exercise 2 ]
5, (Carrying cost proof) Complete the second half of the proof of the "forward price formula with carrying cost" in Section 10 3 To construct the arbitrage, go long one unit of a forward and short one unit spot To execute the short, it is necessary to borrow the asset from someone, say, Mr X. As part of our arrangement with Mr. X we ask that he give us the carrying costs as they would normally occur, since he would have to pay them if we did not borrow the asset We then invest these cash flows At the final time we buy one unit as obligated by our forward and repay Mr. X Show the details of this argument
6 (Foreign currency alternative) Consider the situation of Example 10 10. Rather than shorting a futures contract, the U S firm could borrow 500/(1 ~Hg) Deutsche mark (where tq is the 90day interest rate in Germany), sell these marks into dollars, invest the dollars in Tbills, and then later repay the Deutsche mark loan with the payment received for the German order Discuss how this procedure is related to the original one
7, (A bond forward) A certain 10year bond is currently selling for $920 A friend of yours owns a forward contract on this bond that has a delivery date in 1 year and a delivery price of $940 The bond pays coupons of $80 every 6 months, with one due 6 months from now and another just before maturity of the forward The current interest rates for 6 months and I year (compounded semiannually) are 1% and 8%, respectively (annual rates compounded every 6 months) What is the current value of the forward contract?
8. (Simple formula) Derive the formula (10 6} by converting a cash (low of a bond to that of the fixed portion of the swap
9. (Equity swapo) Mr A Gaylord manages a pension fund and believes that his stock selection ability is excellent However, he is worried because the market could go down He considers entering an equity swap where each quarter /, up to quarter M, he pays counterparty B the previous quarter's total rate of return r, on the S&F 500 index times some notional principal and receives payments at a fixed rate t on the same principal The total rate of return includes dividends Specifically, 1 + /,• = (5/ + where S, and cli are the values of the index at i and the dividends received from / — 1 to i, respectively Derive the value of such a swap by the following steps:
(«) Let Vj~i(Si +di) denote the value at time i — 1 of receiving S( + dt at time i Argue that V5_i(5i +</,•) = .5_i and find
(d) Find the value of the swap
10. (Forward vanilla) The floating rate portion of a plain vanilla interest rate swap with yearly payments and a notional principal of one unit has cash flows at the end of each year defining a stream starting at time 1 of (t{),t[,c;>, , t.vi), where tf is the actual short rate at the beginning of year i Using the concepts of forwards, argue that the value at time zero oi C( to be received at time i + 1 is i/(0, / 4 1)/,, where ij is the short rate for time i implied by the current (time zero) term structure and d(0, / + I) is the implied discount factor to time i + \ The value of the stream is therefore ^iHn' ' + Show that this reduces to the formula for V at the end of Section 10 5
11. (Specific vanilla) Suppose the current term structure of interest rates is ( 070, 073, .077, 081, ,084, .088) A plain vanilla interest rale swap will make payments at the end of each year equal to the floating short rate that was posted at the beginning of that year A 6year swap having a notional principal of $10 million is being configured
(a) What is the value of the floating rate portion of the swap?
(b) What rate of interest for the fixed portion oi the swap would make the two sides of the swap equal?
12. (Derivation) Derive the meanvariance hedge formula given by (10.12)
13. (Grapefruit hedge) Farmer D. Jones has a crop of grapefruit that will be ready for harvest and sale as 150,000 pounds of grapefruit juice in 3 months Jones is worried about possible price changes, so he is considering hedging. There is no futures contract for grapefruit juice, but there is a futures contract for orange juice. His son, Gavin, recently studied minimumvariance hedging and suggests it as a possible approach Currently the spot prices are $1 20 per pound for orange juice and $1 50 per pound for grapefruit juice The standard deviation of the prices of orange juice and grapefruit juice is about 20% per year, and the correlation coefficient between them is about 7 What is the mini mumvariance hedge for farmer Jones, and how effective is this hedge as compared to no hedge?
14. (Opposite hedge variance) Assume that cash flow is given by v — 5y W + (Ft  Fq)Ii Let a\ = var(5/'), aj. = var(F;), and crSi = cov(Sy , F} )
{a) In an equal and opposite hedge, it is taken to be an opposite equivalent dollar value of the hedging instrument Therefore h = —kW, where k is the price ratio between the asset and the hedging instrument. Express the standard deviation of y with the equal and opposite hedge in the form
(b) Apply this to Example 10.12 and compare with the minimumvariance hedge
15, (Immunization as hedging o) A pension fund has just paid some of its liabilities, and as a result of this transaction the fund is no longer fully immunized The fund manager decides that instead of changing the portfolio, the firm should hedge its position using a futures contract on a Treasury bond The fund manager wants to hedge against parallel changes to the spot rate curve Use the following set of information to determine the numerical values of the hedging position:
• Yearly spot rate sequence: 05, 053, 056, 058, 06, 061
o L iabilities: $1 million in Í year, $2 million in 2 years, and $1 million in 3 years 0 Current bond portfolio: $4.253 million in par value of zerocoupon bonds maturing in 2 years. (Use the continuoustime formulas for discounting: e~rr.)
• The hedge is to be constructed using futures contracts on zerocoupon bonds maturing in 6 years, with a contract delivery date in 1 year
16, (Symmetric probability o) Suppose the wealth thai is to be received at a time T in the future has the form where a is a constant and x is a random variable The value of the variable h can be selected by the investor. Suppose that the investor has a utility function that is increasing and strictly concave Suppose also that the probability distribution of a* is symmetric; that is, ..v and —a have the same distribution. It follows that E(a) = 0 and that the investor cannot influence the expected value of wealth
(b) Apply this result to the corn farm problem to show that the optimal futures position is +4,000
17, (Double symmetric probability o) Suppose that revenue has the form where h can be chosen and x and y are random variables The distribution of a and v is symmetric about (0,0); that is, —a, —v has the same distribution as ,v, v Show that the choice of h that minimizes the variance of R is
18, (A general farm problem o) Suppose that, as in the corn farm example, the farm has random production and the final spot price is governed by the same demand function However, the crop of the farm is not perfectly correlated to total demand, but oto and ajy are known The current futures price is also equal to the expected final spot price Show that the minimumvariance hedging position is o = Was x B
Check the solution for the special cases (a) D = 100C and (b) aCD = 0 [Hint: Use Exercise 17 ]
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